Efficient hybrid-symbolic methods for quantum mechanical calculations

Scott, Tony; Zhang, Wenxing

doi:10.1016/j.cpc.2015.02.009

Cited by 9 publications

(9 citation statements)

References 19 publications

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“…As an alternative, we use the hybrid symbolic-methods [38][39][40], in particular the generation of optimized MATLAB code [41] to apply the variational principle. We used MATLAB functions to numerically solve all the integrals of the class in (17) but all other integrals are solved analytically using Maple.…”

Section: Variational Principlementioning

confidence: 99%

Solution of the logarithmic Schrödinger equation with a Coulomb potential

Scott

Shertzer

2018

J. Phys. Commun.

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The nonlinear logarithmic Schrödinger equation (log SE) appears in many branches of fundamental physics, ranging from macroscopic superfluids to quantum gravity. We consider here a model problem, in which the log SE includes an attractive Coulomb interaction. We derive an analytical solution for the ground state energy and wave function as a function of the strength of the logarithmic interaction. We develop an iterative finite element method to solve the Coulombic log SE for the spherically symmetric states. The ground state results agree with the exact solution to better than one part in 10 10 . The excited states (n>1) are converged to better than one part in 10 8. We also construct a remarkably simple variational wave function, consisting of a sum of Gaussons with n free parameters. One can obtain an approximation to the energy and wave function that is in good agreement with the finite element results. Although the Coulomb problem is interesting in its own right, the iterative finite element method and the variational Gausson basis approach can be applied to any central force Hamiltonian.

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Section: Variational Principlementioning

confidence: 99%

Solution of the logarithmic Schrödinger equation with a Coulomb potential

Scott

Shertzer

2018

J. Phys. Commun.

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“…Multi-atomic/multielectronic systems are beyond the realm of obtaining analytical solutions, even for the simples cases of the Hydrogen molecule or the Helium atom [5]. In the context of multi-atomic systems of Quantum Chemistry, numerous techniques exist to approximate the wavefunction - [6] and references therein -that depend on tailoring Physics concepts to code such as the Variational Principle, Ritz Ansatz, etc with additional constraints to model a particular problem. It would desirable to have a generic wavefunction solver suitable for a variety of potentials.…”

Section: Background and Motivationmentioning

confidence: 99%

“…where x d i represents the position of particle i after d th iteration, v d i is the velocity of particle i after d th iteration, P i represents the best position till now for the i th particle, G is the best global position found till now, β is the momentum factor and M d+1 i is the effect of momentum in (d + 1) th iteration. Recursively expanding eq (6), we get -…”

Section: B Conversion To An Optimization Problemmentioning

confidence: 99%

A Swarm Variant for the Schrödinger Solver

Jivani¹,

Vaidya²,

Bhattacharya³

et al. 2021

Preprint

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This paper introduces the application of the Exponentially Averaged Momentum Particle Swarm Optimization (EM-PSO) as a derivative-free optimizer for Neural Networks. It adopts PSO's major advantages such as search space exploration and higher robustness to local minima compared to gradient-descent optimizers such as Adam. Neural network based solvers endowed with gradient optimization are now being used to approximate solutions to Differential Equations. Here, we demonstrate the novelty of EM-PSO in approximating gradients and leveraging the property in solving the Schrödinger equation, for the Particle-in-a-Box problem. We also provide the optimal set of hyper-parameters supported by mathematical proofs, suited for our algorithm 1 .

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“…When considering quantum systems, the following consequence of actions is usually performed [5][6][7][8]:…”

Section: Introductionmentioning

confidence: 99%

A new class of exact solutions of the Schrödinger equation

Perepelkin

Садовников

Иноземцева

et al. 2018

Continuum Mech. Thermodyn.

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The aim of this paper is to find the exact solutions of the Schrödinger equation. As is known, the Schrödinger equation can be reduced to the continuum equation. In this paper, using the non-linear Legendre transform the equation of continuity is linearized. Particular solutions of such a linear equation are found in the paper and an inverse Legendre transform is considered for them with subsequent construction of solutions of the Schrödinger equation. Examples of the classical and quantum systems are considered.

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Efficient hybrid-symbolic methods for quantum mechanical calculations

Cited by 9 publications

References 19 publications

Solution of the logarithmic Schrödinger equation with a Coulomb potential

Solution of the logarithmic Schrödinger equation with a Coulomb potential

A Swarm Variant for the Schrödinger Solver

A new class of exact solutions of the Schrödinger equation

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